Why can't anyone prove that one plus one equals two? This question might sound absurd at first. After all, we learn this basic arithmetic fact in early childhood. However, in mathematics, proving even the simplest statements requires building them from fundamental axioms and definitions. The challenge isn't that one plus one doesn't equal two, but rather that we must first define what we mean by one, by addition, and by two, before we can rigorously prove their relationship.The foundation for proving one plus one equals two lies in the Peano axioms, developed by Italian mathematician Giuseppe Peano in the late nineteenth century. These axioms define natural numbers from the ground up. They start with zero as a natural number, and define a successor function that generates the next number. One is defined as the successor of zero, two is the successor of one, and so on. The axioms ensure that each number has a unique successor, and that zero is not the successor of any number. With these axioms, we can build the entire system of natural numbers and define addition rigorously.Now that we have the Peano axioms, we can define addition. Addition is not a primitive operation, but is constructed recursively from the successor function. We define that any number plus zero equals that number. And we define that a number plus the successor of another number equals the successor of their sum. Using these rules, we can compute any addition. For example, two plus one equals two plus the successor of zero, which by our second rule equals the successor of two plus zero, which simplifies to the successor of two, which is three. This recursive definition allows us to prove addition facts rigorously.Finally, we can prove that one plus one equals two. We start with one plus one. By definition, one is the successor of zero, so we can write this as one plus the successor of zero. Using our recursive definition of addition, this equals the successor of one plus zero. By the first rule of addition, one plus zero equals one. So we have the successor of one. And by definition, the successor of one is two. Therefore, one plus one equals two. This proof may seem elaborate for such a simple fact, but it demonstrates the rigor required in mathematics. Every statement must be built from fundamental axioms and definitions.So why does this matter? The proof of one plus one equals two demonstrates the foundational nature of mathematics. Mathematics is not just a collection of facts, but a logical structure built from the ground up. Every theorem, no matter how complex, ultimately traces back to basic axioms. This rigorous approach ensures that mathematical knowledge is reliable and free from hidden assumptions. When we prove even the simplest statements from first principles, we guarantee that our entire mathematical edifice stands on solid ground. This is why mathematicians can prove that one plus one equals two, and why this proof, though seemingly trivial, represents the essence of mathematical reasoning.